Mapping class groups of surfaces of genus $\ge 3$ do not virtually surject to $\mathbb{Z}$

Abstract

We prove a well known conjecture of Nikolai Ivanov which states that if $X$ is a surface of genus $\ge 3$ (with any number of punctures and boundary components), $\mathrm{Mod}(\mathrm{X})$ is the mapping class group of $X$, and $K<\mathrm{Mod}(\mathrm{X})$ is a finite-index subgroup, then $K$ does not virtually surject to $\mathbb{Z}$. As a corollary of this we get that $H_1(Z;\mathbb{Q})=0$ whenever $Z$ is a finite cover of ${\mathcal{M}}_{g,n}$, the moduli space of complex algebraic curves of genus $g\ge 3$ with $n$ marked points.

arXiv commentThe central problem is that Lemmas 5.3 and 5.4 in the paper (the folding trick and the folding trick for covers) are incorrect. As these lemmas are the heart of the rest of the proof, I am withdrawing the claim. I would like to extend special thanks to Julien Marche, Bram Petri, and Maxime Wolff for their careful reading and comments that lead to the discovery of the problems in the proof

References in corpus (5)

• Quotients of surface groups and homology of finite covers via quantum representations
• Abelian quotients of subgroups of the mapping class group and higher Prym representations
• Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_n)
• C^1 actions of the mapping class group on the circle
• Finite covers of graphs, their primitive homology, and representation theory